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<p>A higher order ODE can be converted into a system of first order equations. For example, for (<a href="" class="xref" data-knowl="./knowl/eq7_1.html" title="Equation 6.1.1">(6.1.1)</a>), we introduce</p>
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\begin{equation*}
x_1=y,\quad x_2=y^{\prime}.
\end{equation*}
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<p class="continuation">Then we may convert the second equation in (<a href="" class="xref" data-knowl="./knowl/eq7_1.html" title="Equation 6.1.1">(6.1.1)</a>) as</p>
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\begin{equation*}
\begin{aligned}
&amp;x_1^{\prime}=x_2,\\
&amp;x_2^{\prime}=f(x, x_1, x_2).
\end{aligned}
\end{equation*}
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<p class="continuation">A system of two first order equations for two unknowns <span class="process-math">\(x_1\)</span> and <span class="process-math">\(x_2\text{.}\)</span></p>
<span class="incontext"><a href="sec6_1.html#p-245" class="internal">in-context</a></span>
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